Evaluation and Simulation of Black-box Arc Models for High-Voltage Circuit-Breakers

(1)

Evaluation and Simulation of Black-box Arc Models

for High Voltage Circuit-breakers

Niklas Gustavsson

LiTH-ISY-EX-3492-2004

(2)

(3)

Evaluation and Simulation of

Black-box Arc Models for High Voltage

Circuit-breakers

Examensarbete utfört i Reglerteknik och

Kommunikationssystem vid Linköpings Tekniska Högskola av

Niklas Gustavsson

LiTH-ISY-EX-3492-2004

Handledare: Markus Gerdin

Examinator: Torkel Glad

Linköping, 19 mars 2004

(4)

(5)

Avdelning, Institution Division, Department

Institutionen för systemteknik

581 83 LINKÖPING

Datum Date 2004-04-02 Språk

Language Rapporttyp Report category ISBN

Svenska/Swedish X Engelska/English

Licentiatavhandling

X Examensarbete ISRN LITH-ISY-EX-3492-2004

C-uppsats D-uppsats Serietitel och serienummer _{Title of series, numbering} ISSN

Övrig rapport

____

URL för elektronisk version

http://www.ep.liu.se/exjobb/isy/2004/3492/

Titel

Title Utvärdering och simulering av black-box ljusbågsmodeller för högspänningsbrytare Evaluation and Simulation of Black-box Arc Models for High-Voltage Circuit-Breakers

Författare

Author Niklas Gustavsson

Sammanfattning

Abstract

The task for this Master thesis was to evaluate different black-box arc models for circuit-breakers with the purpose of finding criteria for the breaking ability. A black-box model is a model that requires no knowledge from the user of the underlying physical processes. Black-box arc models have been used in circuit-breaker development for many years. Arc voltages from tests made in the High Power Laboratory in Ludvika were used for validation, along with the resistance calculated at current zero, R0, and 500 ns before current zero, R500.

Three different arc models were evaluated: Cassie-Mayr, KEMA and an arc model based on power calculations. The third model gave very good results and if the model is developed further, the breaking ability could easily be estimated.

The arc model based on power calculations could be improved by using better approximations of the quantities in the model, and by representing the current better. A further suggestion for the following work is to combine the second arc model tested, the KEMA model, with the model based on power calculations in order to estimate the KEMA model parameters.

(6)

(7)

Abstract

The task for this Master thesis was to evaluate different black-box arc models for circuit-breakers with the purpose of finding criteria for the breaking ability. A black-box model is a model that requires no knowledge from the user of the underlying physical processes. Black-box arc models have been used in circuit-breaker development for many years. Arc voltages from tests made in the High Power Laboratory in Ludvika were used for validation, along with the resistance calculated at current zero, R0, and 500 ns before current zero, R500.

Three different arc models were evaluated: Cassie-Mayr, KEMA and an arc model based on power calculations. The third model gave very good results and if the model is developed further, the breaking ability could easily be estimated.

The arc model based on power calculations could be improved by using better approximations of the quantities in the model, and by representing the current better. A further suggestion for the following work is to combine the second arc model tested, the KEMA model, with the model based on power calculations in order to estimate the KEMA model parameters.

The R0 and R500 values should also be calculated from more tests, in order to find a clear limit of the breaking ability.

(8)

(9)

Acknowledgements

This thesis would have been impossible to accomplish without the help from some people.

First, I wish to thank my supervisors at ABB, Staffan Jacobsson and Fredrik Jansson, who answered all my questions patiently and gave me valuable advice and suggestions during the work.

I would also like to thank my supervisor at the Department of Electrical Engineering at Linköping University, Markus Gerdin, for valuable feedback on the report and for helping me with all the practical issues concerning the thesis.

(10)

(11)

1 INTRODUCTION ...11

1.1 THE ABB GROUP...11

1.1.1 Power Technologies ...11

1.1.2 Automation Technologies ...11

1.1.3 Oil, Gas and Petrochemicals...11

1.1.4 High Voltage Products unit ...11

1.2 CIRCUIT-BREAKERS...12

1.3 TESTING IN THE LABORATORY...13

1.3.1 Test circuit and method ...13

1.3.2 Measurements...13

1.4 BLACK-BOX MODELS...15

2 PROBLEM FORMULATION...16

2.1 TASK...16

2.2 PREREQUISITES...16

3 VALIDATION AND CRITERIA FOR THE BREAKING ABILITY...17

3.1 VALIDATION DATA...17

3.2 BREAKING ABILITY...18

3.2.1 Calculations of the arc resistance ...19

3.2.2 Arc resistance from the validation data...20

4 SOFTWARE...22

4.1 ATP AND ATPDRAW...22

4.2 POSTPROCESSORS...23

4.3 IMPLEMENTATION OF THE CIRCUIT-BREAKER MODEL...23

4.3.1 Model of the circuit-breaker and the arc...23

4.3.2 The test circuit ...24

5 CASSIE-MAYR ARC MODEL...26

5.1 MODEL EQUATIONS...26

5.2 IMPLEMENTATION...28

5.2.1 Transformation into discrete form...28

5.2.2 Calculation of the resistance in the arc model ...28

5.3 MODEL PARAMETERS...29

5.4 MODEL VALIDATION...30

5.4.1 Arc voltage...30

5.4.2 Resistance ...31

5.5 CONCLUSIONS AND COMMENTS...32

(12)

6.4 MODEL VALIDATION...36 6.4.1 Arc voltage...36 6.4.2 Resistance ...38 6.4.3 Comments ...39 6.5 NEW DERIVATIVE APPROXIMATION...39 6.6 MODEL VALIDATION...40 6.6.1 Arc voltage...40 6.6.2 Resistance ...42

7 ARC MODEL USING POWER CALCULATIONS ...43

7.1 BACKGROUND...43

7.2 SIMPLIFICATIONS AND IMPLEMENTATION...44

7.2.1 Speed, length, volume and temperature...44

7.2.2 Area and radius ...44

7.2.3 Pressure...46 7.2.4 Density...48 7.2.5 Specific enthalpy...49 7.2.6 Mass flows ...50 7.3 MODEL EQUATIONS...51 7.4 MODEL VALIDATION...54 7.4.1 Arc voltage...54 7.4.2 Resistance ...56

8 OVERALL CONCLUSIONS AND RECOMMENDATIONS ...60

8.1 COMPARISON OF THE ARC MODELS...60

8.2 RECOMMENDATIONS...60

9 REFERENCES...62

10 APPENDIX...63

10.1 CASSIE-MAYR MODEL PARAMETERS...63

10.2 KEMA MODEL PARAMETERS...63

(13)

Introduction

1 Introduction

This Master thesis has been performed at ABB Power Technologies, High Voltage Products, in Ludvika. The thesis concerns evaluation and

simulation of different arc models for high voltage circuit-breakers. In this section, ABB will be presented along with some background of circuit-breakers and the testing procedure. Secondly, the problem definition is described. The third section is about validation of the models, and in the fourth, the software used will be presented. Then, in the following sections, the arc models are presented along with the conclusions made. Finally, overall conclusions and discussions are made and a reference list is given.

1.1 The ABB group

ABB, Asea Brown Boveri, is a multinational group consisting of the divisions Power Technologies, Automation Technologies and Oil, Gas and Petrochemicals. The group is located in about 100 countries and has approximately 140 000 employees. In Sweden, ABB has about 10 500 employees. The major localities are Västerås with over 5 000 employees and Ludvika with around 2 400. [1]

1.1.1 Power Technologies

Power Technologies serves customers, with industrial and commercial applications, with products and solutions for power transmission and distribution. Power Technologies consists of several units, such as High Voltage Products (see section 1.1.4), Power Systems and Power

Transformers.

1.1.2 Automation Technologies

Automation Technologies deliver solutions for control, motion, protection, and plant integration for process and utility industries.

1.1.3 Oil, Gas and Petrochemicals

Oil, Gas and Petrochemicals supply products, systems and services to the oil- and gas industries.

1.1.4 High Voltage Products unit

High Voltage Products develop, manufacture and sell high voltage products for HVAC1_{- and HVDC}2_{systems. The unit exports up to 90 % of the}

(14)

Introduction

capacitors and cooling systems, where the first four are located in Ludvika and the last in Landskrona.

High Voltage Products also have one of the most modern high power laboratories in the world for testing of the products, the High Power Laboratory, located in Ludvika.

1.2 Circuit-breakers

Circuit-breakers are used in power networks to interrupt the current of the network if a fault appears, e.g. short circuits. They are also used to connect or disconnect parts of the network. Figure 1-1 shows an 80 kA HPL420 circuit-breaker.

Figure 1-1: HPL420 80 kA circuit-breaker. Picture taken from [1]

In simple terms, circuit-breakers consist of a plug that is in connection with a contact when the breaker is closed. The current then flows right through the breaker. To interrupt the current, the plug and the contact is separated with rather high speed, resulting in an electric arc in the contact gap between the plug and the contact. This is illustrated in Figure 1-2.

(15)

Introduction

The arc period is defined as the time from contact separation to

interruption. During the arc period, very high conductivity is desired in the breaker. That is equal to a low power development. When the current reaches zero, the breaker must change from being a good conductor to a good insulator in very short time. If this transition is fast enough, the arc is extinguished and the current will be interrupted and only a small current will flow in the breaker. Otherwise, the arc will re-ignite. As a quenching medium, SF6 gas is used. [2]

1.3 Testing in the laboratory

Here the testing procedure and the measurements in the High Power Laboratory will be explained.

1.3.1 Test circuit and method

Because circuit-breakers must be able to handle large powers, in the gigawatt area, the breakers can not be tested directly. Some sort of synthetic testing must therefore be used. The High Power Laboratory in Ludvika uses the current injection method, which is a widely accepted synthetic test method [2].

The test circuit consists of a current circuit, in which a regular power generator is used to produce full test current through an auxiliary breaker and the test breaker under reduced voltage, and a voltage circuit which includes a charged capacitor bank. The capacitor bank is charged with the test voltage.

Because an alternating current flows in the circuit-breaker, the interruption has to take place at a zero crossing. This zero crossing is called current zero. Right before current zero, the voltage circuit is connected using a spark gap. The capacitor bank is then unloaded and a current pulse is injected and added to the current through the test breaker. This makes the total power very large. The current through the auxiliary breaker is forced to zero and thereby interrupts the current circuit. At the end of the injected current the test breaker has a current zero and a possibility to interrupt the current. By a suitable choice of circuit parameters, a result will be obtained that fully corresponds to the effect from real usage.

1.3.2 Measurements

(16)

Introduction

The most interesting quantities are the voltage across the test circuit-breaker, which is equal to the arc voltage, and the current through this breaker. The injected current is useful for determining if a test is a success or failure. For a successful test, the injected current is just a half-period long, because it has the same current zero as the current from the current circuit. For a failure the injected current is longer. See Figure 1-3 and Figure 1-4 for examples of this.

Figure 1-3: Successful interruption

(17)

Introduction

1.4 Black-box models

In [3], black-box models are defined as a model where only the relation between input and output signals is described, without considering the underlying physical processes. In this thesis, knowledge of the physical processes is required when evaluating and developing the arc models. When used, however, the arc models will be considered black-box models, without requiring any knowledge of the underlying physics.

Black-box models have been used in circuit-breaker development for many years, even before the use of computers. They have proved to be very useful even though they consist of relatively simple relations. By calculating e.g. the conductance or resistance in the contact gap, it is possible to predict

• The circuit-breaker’s influence on the power network.

• The test circuit’s influence on the breaking ability, i.e. the influence on the test results by the laboratory used.

• The breaking ability for different breaking cases.

It is very difficult to calculate the state of a contact gap under the influence of an arc even when using the most advanced calculating tools. Different arc models have been implemented and give good results, but a desire is to simulate the circuit-breaker together with a model of the test circuit used in the laboratory. Therefore, an arc model must be implemented together with the test circuit in a suitable program.

(18)

Problem formulation

2 Problem

formulation

This thesis is meant to give a detailed study of different black-box models with the purpose to evaluate, combine, improve and apply to already existing circuit-breakers. In this section, the task will be defined and the prerequisites will be given.

2.1 Task

• Evaluating known black-box models

During what part of the breaking period are they applicable? How are they constructed?

On which relations and assumptions are they based?

• Combining the most promising models for the large- and small current phases to receive a model for the entire breaking period.

Can the degree of simplifications be decreased?

Can the energy and mass equations from the existing ”physical” models be used?

• Applying one or more of the following cases

Choose a circuit-breaker and a breaking case and make criteria of tests already made.

Estimate the effect of different test circuits for a test case. Estimate the performance of the circuit-breaker in a real network

(same test case as above).

The work will be based on evaluating known arc models and comparing them to validation data. The best model will be compared to criteria for the breaking ability.

2.2 Prerequisites

The arc models are to be implemented using a software package called ATP. An existing arc model implemented in ATP was available, see [4]. The arc model was the Cassie-Mayr model presented in section 5. Also some test circuits without any arc models were implemented [5]. This combination served as a starting point for the continuing work.

(19)

Validation and criteria for the breaking ability

3 Validation and criteria for the breaking ability

Here the validation data used will be presented together with a discussion of the breaking ability.

3.1 Validation data

Data from tests made at the High Power Laboratory in Ludvika were used as validation data. The tests were made on the same test circuit and with the same breaker. The breaker is a 145 kV, 40 kA, 60 Hz

circuit-breaker. The test circuit is called 1346-1 [6]. A drawback with this data is that the circuit-breaker used in the tests mostly managed to interrupt the current, resulting in few examples of failed tests. Nine test shots were available, and in only one of these the breaker failed to interrupt. The test shots are numbered from 670551 to 670559. All were successful

interruptions except the last, 670559, which was a failure.

The quantities measured in these tests are described in section 1.3.2. The most useful for validation is the arc voltage. By comparing it to the calculations made using the arc model, the calculated voltage could be calibrated to fit the measured one, and be used as a measure of the quality of the arc models.

The tests were made under Short Line Fault conditions. Short Line Fault is a type of fault that appears for instance when a tree has fallen over the line some kilometres from the circuit-breaker, and is one of the breaking cases that the breaker must be able to handle.

The current through the circuit-breaker will be used as input parameters to all the arc models. It is therefore important to represent it well. In Figure 3-1 and Figure 3-2, the current is plotted for two of the test shots used as validation data. The injected current is shown as a small bump at current zero for a successful interruption, and with a bump also on the negative side for the failed interruption.

The amplitude and phase of the current differs between the test shots, but current zero occurs at the point t = 25 ms for all test shots. Because the arc models will only be evaluated during the arc period, good agreement for the calculations is only required up to this point.

(20)

Figure 3-1: Measured current, successful interruption

Figure 3-2: Measured current, failed interruption

3.2 Breaking ability

The breaking ability depends on many factors. It has been investigated [7] that the conductance of the arc at current zero is a good electrical parameter for determination of the ability of the circuit-breaker to interrupt the

current. The reason for this is that large power loss increases the chance for interruption. Large power loss is equal to small conductance. Because the resistance is the inverse of the conductance, the resistance at current zero is an equally good parameter.

(21)

takes place. A major task will be to make accurate calculations of the resistance at current zero for different test cases.

3.2.1 Calculations of the arc resistance

Generally, a resistance can be calculated using Ohm’s law. That is

i

u

R

i

R

u

=

⋅

⇔

=

A time derivation of this formula results in the following expression

i

u

R

i

R

u

′

=

⇔

′

⋅

=

′

The arc resistance at current zero, R0, can therefore be calculated as

























=

→ →

)

(

lim

)

(

lim

)

(

)

(

)

(

0

t

i

dt

d

t

u

dt

d

t

i

t

u

t

R

CZ t arc CZ t CZ CZ arc CZ arc

where uarc is the arc voltage, and i is the current through the circuit-breaker. CZ stands for current zero. The derivatives can be re-written in a shorter form:

dt

du

t

u

dt

d

arc arc CZ t





=









→

(

)

&

lim

dt

di

t

i

dt

d

CZ t





=









→

(

)

&

lim

where

du

_arc

dt

is the voltage steepness at current zero and

di

dt

is the steepness of the current.

It can be difficult determining the voltage steepness because of rapid changes at current zero. The resistance 500 ns before current zero, R500, is an equally good measure of the breaking ability as the current zero resistance. It has been found in earlier studies that there is a clear limit of the R0 and R500 values between successful and failed interruptions.

(22)

The area around current zero of the arc voltage and current are plotted in Figure 3-3. These plots show the reason for calculating the voltage steepness in a very short time interval. It is also shown that the peak of the arc voltage has a deep impact on the breaking ability. The current behaves almost linearly around current zero, so a longer time interval can be used for calculating the current steepness.

Figure 3-3: Measured current and arc voltage at current zero

3.2.2 Arc resistance from the validation data

Both the arc resistance at current zero, R0, and 500 ns before, R500, has been calculated from the validation data. These values are found in Table 1 and are also plotted in Figure 3-4 and Figure 3-5. There are no major

differences between the R0 and R500 values, which means both are equally good for validation.

It is almost impossible finding a clear limit between successful and failed interruptions from these values, but a tendency is shown. There should not be a clear limit between test shots from the same test series either, because the shots are rather similar to each other. To get a fair view of the breaking limit, a lot more values must be calculated from test series with as many failed as successful interruptions.

(23)

Table 1: Calculated value of R0 and R500 from the validation data

Test shot R0 [Ω] R500 [Ω] 670551 332.66 341.06 670552 352.61 346.68 670553 347.87 323.46 670554 347.30 355.82 670555 358.01 368.01 670556 336.22 328.86 670557 320.99 345.73 670558 339.18 373.18 670559 (failure) 327.93 336.14

(24)

Software

4 Software

In this section, there is a short description of the software used. The

implementation of the arc models and the test circuit will also be described.

4.1 ATP and ATPDraw

ATP, Alternative Transients Program, is a universal program for digital simulation of electric power systems. ATP was developed in 1984 out of the Electromagnetic Transients Program, EMTP. EMTP was at first a public domain software, but when it was to be released commercially, ATP was developed as a free version of EMTP. ATP requires a license, but licensing is free of all charge.

ATPDraw is a graphical preprocessor to ATP. Using the mouse, the user can construct a model of the circuit to be simulated. This is shown in Figure 4-1. ATPDraw then creates the input file to ATP for simulation. The output files from ATP are either binary .pl4 files or ASCII-text.

Figure 4-1: ATPDraw

(25)

Software

looks. The user can also create own components by using so called MODELS-objects with its code written as a text file. [8], [9]

The major drawback with ATP is that it is not possible handling variable timesteps. It can sometimes be useful having a longer timestep for static parts of the simulation and a smaller timestep for more dynamic parts. Instead, a small timestep must be used for the whole simulation resulting in very large output files, or a longer timestep can be used with the risk of simulating an incorrect behaviour. Mostly, a compromise must be made. Another drawback is that it is difficult to handle derivatives in the MODELS-code. Regular time-derivatives can easily be calculated, and linear time-dependent differential equations can also be solved. More general derivatives must, however, be implemented manually by re-writing into recursive form using some derivative approximations.

4.2 Postprocessors

A postprocessor is a software used to present the calculated data in a suitable way, e.g. as a plot. Different programs were used as

postprocessors, depending on the purpose. PlotXY works well with the .pl4 output files from ATP. It is fast and has a good zoom function. To be able to compare calculated data with measured, ADAMS Postprocessor was used. It can import ascii-data from different files and plot in the same window to make easy comparisons between calculations and

measurements. The program can also make rather advanced calculations with the imported data.

4.3 Implementation of the circuit-breaker model

In this section, there is a description of the ATPDraw implementation of the circuit-breaker and the test circuit. These implementations were already available at the beginning of the thesis. The circuit-breaker implementation is the same for all arc models, only the arc model differs. The test circuit implementation is also the same for all arc models.

4.3.1 Model of the circuit-breaker and the arc

An existing model of the circuit-breaker in ATPDraw was used, shown in Figure 4-2. The different arc models are implemented as MODELS objects. When using different arc models, different MODELS-objects are used together with some model parameters that are depending on the present model and test shot.

(26)

Software

the one representing the spark gap. The calculated resistance of the arc, placed parallel to the switch, is the output signal.

Figure 4-2: ATP breaker

When the switch is closed, the current flows right through it, and the MODELS-code is not used. The circuit-breaker is then symbolised by a very small resistance, about 10-5_{Ω. When the switch is opened, the current} flows through the MODELS-controlled resistance, which represents the arc. The size of the resistance is calculated by the model equations and gives rise to an arc voltage. The opening time is chosen so that the expected arcing time is obtained. After successful interruption the resistance

becomes very large, which symbolises that the arc is extinguished. 4.3.2 The test circuit

The arc model is used in an ATPDraw-model of the synthetic circuit used in the High Power Laboratory. The ATP model of the test circuit is shown in Figure 4-3. The breaker to be tested is represented by the ATP breaker model. A time-controlled switch represents the spark gap in the voltage circuit. The values of the components in the circuit were obtained from [6].

(27)

Software

Figure 4-3: Test circuit in ATP

As shown in Figure 3-1 and Figure 3-2 in the previous section, the current from real tests is not always symmetrically sinus-shaped. Therefore an exponential current source component was added to the circuit to represent the DC-part of the current. In Figure 4-4, the calculated current is plotted together with the measured for one of the test shots.

(28)

Cassie-Mayr arc model

5 Cassie-Mayr arc model

In this section, the Cassie-Mayr arc model will be explained. This model was already implemented, but some work has been done on the model parameters. First, a brief background of the model is given and then the implementation, model parameters and results are presented. Finally, some conclusions are made.

5.1 Model equations

The model is based on two equations for calculation of the conductance in an arc, derived by Cassie and Mayr around 1950. A brief description of the derivation of these equations is found in [2] and will be outlined in this section to show the assumptions that the model is based on.

The power of the arc can be calculated as

dt

dQ

P

i

u

P

=

⋅

=

_loss

+

where u is the arc voltage, i is the current through the breaker, Ploss is the power loss of the arc and Q is the heat content. Further, the conductance G is a function of Q:

( )

=

(

_∫

(

−

)

=

G

Q

G

P

dt

G

_loss

After differentiation the following expression is obtained:

(

loss

)

(

P

loss

)

Q

G

Q

G

dt

dG

G

P

dQ

Q

dG

dt

dQ

Q

dG

dt

dG

−

′

=

⇔

−

=

⋅

=

)

(

)

(

1 )

(

)

(

To solve this equation, Ploss and G(Q) must be known. Cassie and Mayr proposed models for solving the equation.

Some simple relations using Ohm’s law will be used in the derivation of the models:

R

G

I

R

U

I

U

P

=

⋅

=

⋅

=

1

(29)

Cassie assumed that only convection3_{causes the power losses, which} means that the temperature in the arc is constant. This implies that the cross-section area of the arc, A, is proportional to the current and that the voltage over the arc is constant. Using these assumptions, a linear relation between G and Q is obtained. Further,

P

_loss

=

U

_c2

⋅

G

_{, where U}

c is the arc voltage. If these relations are inserted in the equation above, with

G

u

P

=

2

⋅

_{, the following equation is obtained:}













−

=













−

⋅

=

1

2 2 2 2 2 c c c c

U

u

U

u

Q

G

U

dt

dG

G

τ

τc is the time-constant of the conductance because of change in cross-section of the arc. By multiplying both sides of the equation with G, the final expression is obtained, where index c stands for Cassie:













−

⋅

=

_c c c c

_G

U

u

i

dt

dG

2

1 τ

Mayr assumed power losses are caused by thermal conduction at small currents. This means that the conductance is strongly temperature-dependent but fairly intemperature-dependent of the cross-section area of the arc. The area is therefore assumed constant. For constant specific heat capacity,

T

const

Q

=

.

⋅

, where T is the temperature. The electrical conductivity, σ, can be expressed as

σ

=

const

.

⋅

e

T, and

( )

.

Q0

Q

e

const

Q

G

=

⋅

. Mayr

further assumed that the relation between current and voltage is constant and that Ploss = constant = P0. The following equation was then obtained:













−

=













−

=

1

0 0 0 0

P

Q

P

dt

dG

G

τ

_m

τm is the time-constant because of change in temperature without power input in the arc. Both sides of the equation are multiplied with G to obtain the final expression below. Index m stands for Mayr.













−

=

_m m m

_G

P

i

dt

dG

0 2

1 τ

(30)

5.2 Implementation

Only Cassie’s equation can be used for large currents and only Mayr’s around current zero, but that can cause problems at the transition between the models. Hence it is better to use both equations at the same time and series-connect the calculated conductances. For large currents almost all voltage is at the Cassie part and just before current zero the Mayr part takes over [10]. The total conductance is calculated as:

c m

G

1

1 +

=

5.2.1 Transformation into discrete form

To be able to solve the differential equation numerically, the equations are re-written into discrete form using the Euler forward approximation.













∆

−

+

⋅

∆

=

⇔













−

=

−

+ + + m n m n m n m n m n m n n n m n m

t

G

P

i

t

G

P

i

t

G

τ

1

, 0 2 1 , , 0 2 1 , 1 ,













∆

−

+

⋅

∆

=

⇔













−

⋅

=

−

+ + + c n c c n n c n c n c c n n c n n n c n c

t

G

U

u

i

t

G

U

u

i

t

G

τ

1

, 2 1 , , 2 1 , 1 ,

These equations are the basis of the arc model. The resistance of the arc is then calculated as the inverse of the conductance.

5.2.2 Calculation of the resistance in the arc model

The resistances of the Cassie and Mayr parts of the model are calculated separately and summed together as seen below:

m c tot m m c c

R

G

R

G

R

+

=

+

=

+

=

− − 8 8

10

1

10

1

The addition of 10-8_{in the denominator is to avoid division by zero. It also} provides a fixed, large value for the resistance when the conductance gets

(31)

5.3 Model parameters

Because the Cassie-Mayr model already was implemented in ATPDraw, there existed some parameters that could be seen as a starting point of the investigation. Obviously, the parameters vary according to the type of breaker and test case. It was still possible to get a good understanding of how the model works.

Using the original values of the time-constants τm and τc gave rather good results without any significant numerical errors. There were no problems using a smaller timestep either. Hence, these values were used in all tests with this model and no further work was made on determining other values of these parameters.

Uc decides the size of the calculated arc voltage, which is constant when using the Cassie-Mayr model. This is a major drawback, because the arc voltage should be calculated by the model, and not set by an input parameter.

P0 represents the power loss of the arc, and controls the breaking ability. The breaking ability increases with larger power loss. By gradually decreasing P0, a limit for the breaking ability can be found. In Figure 5-1, the smallest possible integer value of P0 before failure is plotted versus different arcing times. It shows that P0 is virtually independent of the arcing time. The breaking ability should vary with the arcing time though, so it is difficult to estimate P0 for different test shots. P0 is therefore not very suitable as a parameter controlling the breaking ability

(32)

5.4 Model validation

5.4.1 Arc voltage

In Figure 5-2 and Figure 5-3, the calculated arc voltage for two of the test shots, 670551 and 670559, is plotted together with the measured. The model parameters used for these calculations are found in Table 4 in the appendix. The value of the parameter P0 has been set to represent a successful interruption for 670551, and to a failure for 670559.

The arc voltage from the Cassie-Mayr model is constant during the whole arc period and the model parameter Uc was used to set its level. Because of this, the same curves were obtained when using a smaller timestep. The voltage was set as accurate as possible for the final half-period, in order to obtain good values of the resistance at current zero.

(33)

Figure 5-3: Arc voltage for 670559 using the Cassie-Mayr arc model

5.4.2 Resistance

The calculated arc resistance for the test shot 670553 is plotted in Figure 5-4. The resistance looks almost the same for all test shots. It is shown in the plot that the resistance reaches a constant level, around 108_{Ω, after} current zero as described in section 5.2.2. Prior to current zero, the resistance is rather small, in the order 10-2_{Ω. This is a good illustration of} the circuit-breakers transition from a good conductor to a good insulator. The large peaks after current zero are probably caused by numerical errors. Because a value of the model parameter P0 that matches the actual test shots can not be found, it is impossible to make a fair comparison between the R0 and R500 values from calculations and validation data.

(34)

5.5 Conclusions and comments

As shown in Figure 5-2 and Figure 5-3, the Cassie-Mayr arc model produces fairly accurate arc voltages. A major drawback is, as pointed out earlier, that the arc voltage is constant and must be set by the user. The R0 and R500 values can not be calculated without good knowledge of the parameter P0, which makes it difficult finding breaking criteria.

The model is very difficult to use as a black-box model, because knowledge about actual measurements is required for each test. The goal is to use only the current and arcing time as input parameters for the model, but the values of the parameters Uc and P0 must be estimated in order to obtain a good approximation of an unknown circuit-breaker’s behaviour.

The model could probably be improved significantly by using non-constant parameters, but the parameters must still be estimated in order to use the model as a black-box model.

(35)

KEMA arc model

6 KEMA arc model

To improve the calculations and to get more realistic results, a more advanced arc model will be implemented, the KEMA arc model. There will be descriptions of the background of the model, implementation and model parameters. The results obtained will be presented along with some

conclusions and comments.

6.1 Background

The KEMA model is described in [7], and an outline of that description is given in this section.

The KEMA arc model is based on the classical equations of Mayr and Cassie described in section 5. It consists of three modified Mayr models in series, representing three sections of the arc. The parameters of each submodel are a time constant Ti [s], a quantity Πi [Aλ-1V3-λ] related to power loss and a dimensionless model parameter λi.

The model equation is:

i i i i i i i

_G

T

U

G

T

dt

dG

1

_i 2

₋

1 Π

=

λ _{i = 1, 2 and 3}

where Gi is the conductance and Ui is the arc voltage for the i:th submodel. If λ = 2, the model equation becomes the classical Mayr model with Π = P0 [W] as the power loss constant. If λ = 1, the equation is the Cassie model and Π = U02, with U0 [V] as the initial arc voltage.

The current I, voltage U and conductance G are described by the following equations:

∑

=

3 1

1

i

G

i

G

∑

=

3 1 i i

U

G

U

I

=

⋅

This model contains nine parameters, but six of them are empirical constants according to the following values and relationships:

(36)

KEMA arc model 1 1 2

T

k

T

=

2 2 3

T

k

T

=

2 3 3

=

Π

k

The first submodel is chosen to be a Cassie-Mayr type arc, the second almost a pure Mayr arc, and the third is a pure Mayr arc. The values of λi remain the same for all tests, while the breaker parameters k1, k2 and k3 depend on the actual circuit-breaker design. They keep their values during all tests on the same circuit-breaker. T1, Π1 and Π2 are considered as free parameters, describing the state of the circuit-breaker. These parameters have a wide range of values and depend on the actual test conditions.

6.2 Implementation

As for the combined Cassie-Mayr arc model, the KEMA model will be re-written into discrete form using the Euler forward approximation. The arc voltage for the different arc sections of the model will be calculated as:

i b i

G

i

U =

where ib is the current flowing through the breaker. Further, the conductances are calculated for each arc section according to:

1 , 2 1 , , −

1 

−











₋

∆

+

Π

∆

=

_i_n i i n i i i n i

G

T

t

U

G

T

t

G

λi

where index n denotes the current time sample and i = 1, 2 and 3 depending on the submodel used.

The total conductance is then calculated the same way as with the Cassie-Mayr model: 8 3 8 2 8 1

10

1

10

1

10

1

− − −

+

₊

+

₊

+

=

G

The resistance of the arc is then calculated as the inverse of the conductance.

6.3 Model parameters

In this section, the different types of model parameters will be explained along with their influence on the calculations.

(37)

KEMA arc model

6.3.1 Circuit-breaker parameters

The parameters k1, k2 and k3 describe the circuit-breaker, and they are the same on all tests with the same breaker. These parameters are used along with the state parameters to calculate T2, T3 and Π3.

In [7], the breaker parameter values are given for three types of circuit-breakers: 72 kV, 123 kV and 145 kV. For a 145 kV circuit-breaker, which is studied in this thesis, the values were

k1 = 5.7 k2 = 10 k3 = 100

These values were used as a start and gave good results. To get different time constants other values were also used, but there were no significant improvements of the calculations. Because of the good results obtained using the values above, these were used in all the following calculations. 6.3.2 Free parameters

The free parameters of the KEMA arc model are T1, Π1 and Π2. T1 is the time constant of the first arc equation and the other time constants depend on T1 as shown in section 6.1. In the model equations, the quotient between the simulation timestep and the time constant is always used. Therefore, there is a strong dependence between the calculated conductance and this quotient.

The parameters Π1, Π2 and Π3 are related to power loss. Because submodel 1 is a Cassie-Mayr type model, which works well for large currents, the parameter Π1 controls the size of the arc voltage. Submodels 2 and 3 are Mayr type models and are better for small currents. Π2 and Π3 therefore determine the breaking ability, which increases with larger power loss. Π3 is calculated directly from Π2, so by gradually decreasing Π2, the limit of the breaking ability can be obtained. A plot of the smallest possible value of Π2 before re-ignition versus arcing time is shown in Figure 6-1. There is not a clear connection between Π2 and arcing time, which means it is difficult finding a suitable value for an unknown test shot. Since the values of Π1 are rather large, and the values of Π2 are small, these parameters do not affect each other that much.

(38)

KEMA arc model

Figure 6-1: Π2 versus arcing time

There were some difficulties finding a value of T1 that gave good results without numerical errors like oscillations and other strange behaviour. The value had to be within a certain interval depending on the simulation timestep. This is probably because three submodels are used and T1, together with the circuit-breaker parameters, determines the time-constant of all of them. The model parameters used for the calculations are found in Table 5 in the appendix.

6.4.1 Arc voltage

The calculated and measured arc voltages for three of the test shots, 670551, 670553 and 670559, are plotted in Figure 6-2 to Figure 6-4. The parameter Π1 was used to calibrate the calculated voltage to fit the measured one for the final half-period. Π2 was set to the limit between successful and failed interruption. For 670551 and 670553 the parameter was set to a successful interruption, and for 670559 to a failure.

(39)

KEMA arc model

Figure 6-2: Arc voltage for 670551 using the KEMA arc model

(40)

KEMA arc model

Figure 6-4: Arc voltage for 670559 using the KEMA arc model

As shown in the plots, the KEMA model gives accurate results for the final half-period when Π1 is given a suitable value. For the earlier parts, the calculated arc voltage is a bit too high, but it is the final part that determines the resistance at current zero.

6.4.2 Resistance

The arc resistance for the test shot 670551 is plotted in Figure 6-5. The plot shows the same behaviour as for the Cassie-Mayr model, with a fast transition from conductor to insulator at current zero.

(41)

KEMA arc model

zero, and could be solved using a smaller timestep. That requires even more work on T1 though.

6.4.3 Comments

The advantage with the KEMA model over the combined Cassie-Mayr model is that the model calculates the arc voltage itself, and is not

depending totally on an input parameter. That implies that the arc voltage is determined more realistically and the result is a quantity that gives good agreements with measured values from the laboratory. The arc resistance also behaves realistically.

Quite a lot of work has to be done with determining proper values for T1, and the arc resistance calculated by the model oscillates heavily at current zero even for some otherwise proper values of T1. The oscillations occur because of numerical errors when the current is approaching zero. A smaller simulation timestep is therefore preferred. The present arc model implementation requires even more work on determining the time constants when using a smaller timestep though.

Even a smaller timestep can cause numerical problems if it brings the calculations outside the region of stability for the numerical method used. Hence, a more advanced derivative approximation has to be used.

6.5 New derivative approximation

To be able to handle smaller timesteps easier, another derivative

approximation will be used. The conductances are now calculated using a fourth-order Runge-Kutta derivative approximation according to the formulas below:

(

)

II n i i i n i i i III I n i i i n i i i II n i i i n i i i I IV III II I n i n i

k

G

T

t

U

G

T

t

k

G

T

t

U

G

T

t

k

G

T

t

U

G

T

t

k

G

i i i













₊

∆

−

Π

∆

=













₊

∆

−

Π

∆

=

∆

−

Π

∆

=

+

=

+ , 2 , , 2 , , 2 , , 1 ,

2

6

1

λ λ λ

(42)

KEMA arc model

Index n denotes the present time sample and i = 1, 2 and 3 depending on submodel used. The total conductance is calculated as earlier and the resistance is obtained by taking the inverse of the conductance.

Here, it is easier finding a value of T1 that works well with the test shots. A smaller timestep was also used here. Parameter settings for these

calculations are found in Table 6 in the appendix. The calculated resistance is equal as the one calculated using Euler forward as shown in Figure 6-5. 6.6.1 Arc voltage

The calculated arc voltages for three of the test shots are plotted in Figure 6-6 to Figure 6-8 together with the measured.

(43)

KEMA arc model

Figure 6-7: Arc voltage for 670553 using KEMA and Runge-Kutta

Figure 6-8: Arc voltage for 670559 using KEMA and Runge-Kutta

When using the Runge-Kutta method and a smaller timestep with the KEMA model, the calculated arc voltage is a little less accurate compared with the Euler forward method. This is illustrated for the shot 670559 in Figure 6-9. The reason for this is unknown. The calculation time is also longer when using Runge-Kutta but these disadvantages can be

compensated by the fact that it is easier finding well-functioning values of T1.

(44)

KEMA arc model

Figure 6-9: Arc voltage for 670559, different timesteps

6.6.2 Resistance

The calculated resistance behaves in the same way as when using the Euler forward method. As for the Cassie-Mayr model, it is very hard finding values of the model parameter Π2 so the R0 and R500 values from the calculations matches the values from the actual test shots.

6.7 Conclusions and comments

When using the Runge-Kutta method, simulation timesteps of 2⋅10-7_and smaller can be used, compared to timesteps of 5⋅10-7_{when using Euler} forward. The oscillations at current zero of the resistance have disappeared. It is also rather easy finding a well-functioning value of T1.

Because estimating the values of the model parameters requires a lot of knowledge of the actual test shots, the KEMA arc model is difficult to use as a black-box model. To be able to predict the ability of the circuit-breaker to interrupt the current, i.e. to calculate R0 and R500, the value of Π2 and Π3 must be determined thoroughly depending on the different test shots. To use this model properly, some more advanced calculations of the parameters must be implemented. One suggestion is to calculate the parameters by using a parameter estimation method together with MatLab4. Another method could be to use time-dependent model parameters, and calculate their values using physical equations.

(45)

Arc model using power calculations

7 Arc model using power calculations

To be able to get even more realistic values of the arc voltage and resistance, a new and more complex model will be implemented. This model is based on physical calculations of the power of the arc. Hopefully this will improve the results without having to put so much work in determining model constants for each test shot.

7.1 Background

The new model is based on equations from an arc model included in the Breaker Simulation Toolkit in a software package called ADAMS5_{. This} model approximates the arc as a cylindrical volume, and takes different flow areas, mass flow, power losses and gas states into consideration [11]. Only the power equations of that model, with some simplifications, will be used here.

The total power of the arc is calculated as:

net mfl ax tur rad tot

q

P

=

+

−

+

where qrad is radiative power loss because of particles hitting the walls inside the breaker and qtur is turbulent power loss from turbulence at the surface of the arc. qax is power loss carried by axial mass flow and qmfl is power loss carried by radial mass flow. qnet is the net power of the arc. In the ADAMS-model, these quantities are calculated as:

(

)

(

arc arc arc

)

arc arc net av rad mfl arc ax ax sound arc arc av arc tur B arc arc rad

V

p

h

V

dt

d

q

h

m

q

h

m

q

v

r

l

h

q

T

r

l

q

⋅

−

⋅

=

⋅

=

⋅

=

⋅

−

=

⋅

=

&

ρ

σ

π

2

4 where l = arc length rarc = arc radius Tarc = arc temperature

(46)

ρarc = mass density in arc vsound = speed of sound in arc max = axial mass content mrad = radial mass content parc = arc pressure

Varc = arc volume

7.2 Simplifications and implementation

The main idea is to calculate the total power of the arc and, by using Ohm’s law, calculate the arc resistance according to the equation below:

2

i

P

R

tot

arc

=

The current through the circuit-breaker will be used as input parameter. In order to use the equations with ATP, some simplifications have to be made. Many of the quantities in the model are current dependent, which is a good reason for using the current as input parameter. To be able to find correct relations between quantities from the ADAMS model and the ATP model, ADAMS-calculations from three of the test shots used as validation data are available for comparison. The shots are 670551, 670553 and 670559. The goal is to find a model that is as simple as possible, but still gives better results than the previous black-box models. Most of the numerical constants used in the model are the same for all test shots. These constants are found in Table 7 in the appendix. In all plots in this section with comparison between ADAMS and ATP calculations, the solid curve is ADAMS calculation and the dashed one is from ATP.

7.2.1 Speed, length, volume and temperature

The calculation of the arc time starts at contact separation and proceeds with the simulation timestep. Both the speed of sound and the contact speed are approximated as constant. The length of the arc is then calculated as

arc

contact

t

v

l

=

⋅

but limited to a maximum value of 0.068 m, the maximum arc length allowed. The arc is approximated as a cylinder, which means that the volume is calculated as the product between length and cross-section area. The temperature can be seen as constant during the whole arc period. 7.2.2 Area and radius

In Figure 7-1, the cross-section area of the arc is plotted against the magnitude of the current for the ADAMS calculations. This shows that

(47)

Figure 7-1: Arc area versus current from ADAMS

Because of the wear in the circuit-breaker, the proportionality constant changes after each test. Using the ADAMS-calculations, the constant can be found for three of the test shots. The accumulated energy, i.e. the sum of the energy of the arc for all the previous shot, can be used as a measure of the wear. The proportionality constant is plotted against the accumulated energy in Figure 7-2. By calculating the accumulated energy for each test shot and comparing to the plot, the value of the proportionality constant, kA, can be obtained. The values of kA for the test shots are found in Table 8 in the appendix.

(48)

The area can now be calculated as

A

_arc

=

k

_A

⋅

i

. A comparison of the areas from ADAMS and ATP are found in Figure 7-3. Because the arc is

cylindrical, the radius of the arc is calculated from the area as

π

arc

A

r

=

.

Figure 7-3: Comparison of area

7.2.3 Pressure

The pressure of the arc is current-dependent, and also depends on the relative length of the arc,

max

l

. As a first approach, the pressure can

therefore be calculated as max 0

_l

l

i

k

p

_arc

=

+

_p

⋅

.

In Figure 7-4, the pressure is plotted against the product of current and relative arc length for the ADAMS calculations. It is shown that there is not a linear relation between these quantities. The last part of the pressure is descending according to an exponential relation,

e

−tτ. There is, however, not a big difference between calculating the pressure using the current-dependent equation above all the way and using the exponential relation for the last part. The model also becomes more complex with the exponential relation, because it is difficult finding the point when the calculation of the

(49)

Figure 7-4: Pressure versus current and length from ADAMS

As for the area, the proportionality constant, kp, changes with the wear in the circuit-breaker. Values of the constant for three of the test shots were found by comparing to ADAMS-calculations and plotted against the accumulated energy. The other values of the proportionality constant can be obtained from Figure 7-5 by calculating the accumulated energy. The values of kp for the test shots are found in Table 8 in the appendix.

(50)

calculations as a start. If the results are bad, the more advanced exponential relation will be used in the calculation.

Figure 7-6: Comparison of pressure

7.2.4 Density

At temperatures over 2000 K, the mass density of the arc is independent of the temperature. The density is, however, proportional to the magnitude of the pressure, as shown in Figure 7-7 below where the density has been plotted against pressure for the ADAMS calculations. The proportionality constant can be found by comparison of plots. The density is calculated as

arc

=

k

ρ

⋅

p

ρ

. Figure 7-8 shows that this relation gives rather good agreement between the calculations, and no more work will be made on this quantity.

(51)

Figure 7-7: Density versus pressure from ADAMS

Figure 7-8: Comparison of density

7.2.5 Specific enthalpy

The specific enthalpy in the arc volume, hav, can be regarded as constant during the arc period. The specific enthalpy of the arc, harc, is related to the pressure, as shown in Figure 7-9, where the ADAMS calculations of the specific enthalpy is plotted against the pressure. As an approach, a simplified approximation, a linear relation, will be used. The enthalpy is

⋅

−

=

(52)

Figure 7-9: Enthalpy in arc versus pressure from ADAMS

In Figure 7-10 below, the calculated harc from ADAMS and ATP are compared. The agreement is not good, but because the total power depends on many quantities, no more advanced approximations should be needed for now. The agreement can be improved by, for instance, using a piece-wise linear relation instead.

Figure 7-10: Comparison of enthalpy

7.2.6 Mass flows

Both the radial and axial mass flows must be calculated. The axial mass flow can be simplified as

m

_&

=

A

⋅

ρ

⋅

v

. The radial mass flow

(53)

content in the arc. marc is then calculated as

m

arc

=

ρ

arc

⋅

V

arc, and the

derivation is performed numerically by ATP.

7.3 Model equations

Using the equations above in the power loss equations, the following model equations are obtained:

(

)

(

)

(

)













_⋅

₋

_⋅

⋅

=

⋅













_⋅

₊

_⋅

⋅

=

⋅

=

⋅

−

⋅

=

⋅

=

arc arc arc arc arc arc arc net net av sound arc arc arc arc mfl mfl arc sound arc arc ax ax sound arc arc av arc tur tur arc arc rad rad

A

t

p

h

A

t

dt

d

v

k

q

h

v

A

l

A

dt

d

k

q

h

v

A

k

q

v

r

l

h

k

q

l

r

T

k

q

&

ρ

4

where krad, ktur, kax, kmfl and knet are calibration constants. These constants are used for compensating the simplifications made. The values of these constants are the same for all test shots and are found in

Table 9 in the appendix. In the following figures, the different power quantities and the total power are compared to the ones in the ADAMS model. The solid lines are the ADAMS quantities and the dashed ones are from ATP. These plots shows that the results obtained using the

simplifications above are very good. There is no need for more complex calculations of these quantities. Since there are only time-derivatives in these equations, there is no need for any derivative approximations either. That is an advantage compared to the previous arc models.

(54)

Figure 7-12: Comparison of turbulent power loss

(55)

Figure 7-14: Comparison of power loss from radial power flow

(56)

Figure 7-16: Comparison of total power

As described earlier, different sets of proportionality constants were used for the different test shots in order to obtain as good correspondence as possible between measured and calculated quantities. The results are presented in the following subsections. A drawback is that the model considers the test shot 670559 as a successful interruption although it should be a failure.

The arc model is rather easy to use. Only the proportionality constants for the area and pressure must be estimated before simulation, and it is easily done. Knowledge of previous test shots is required though, if the wear in the circuit-breaker is to be taken into account.

7.4.1 Arc voltage

It is shown in the following figures that this model gives accurate results for the whole arc period. Even for 670559, which is supposed to be a failure, the calculated arc voltage follows the measured one. This is because the measured arc voltages behave in the same way until current zero for all test shots. Therefore, the resistance at current zero should be calculated equally well as for the other test shots.

(57)

Figure 7-17: Arc voltage for 670551 using power calculations

(58)

Figure 7-19: Arc voltage for 670559 using power calculations

7.4.2 Resistance

The calculated arc resistance for the test shot 670554 is plotted in Figure 7-20. It is showing a more oscillating behaviour after current zero than for the other arc models. This is probably because the arc model is a stiff system, which means all the quantities in the model equations are calculated directly without any differential equations. If some of the equations were differentiated, the oscillations could disappear. However, the transition from a conductor to an insulator is clearly visible.